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Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace
The proving of theorem 3. Let 2A1 > 1. We Make sure that the multi-valued mapping D (t ),0< t < T is weakly invariant. For all ||z0 (■) eF(0) we choose the control u(t,x) = 0, t e[0,T], x е5П, i. e.
I u(t,s)(pk (s)ds = 0, k = 1,2,.... Then,
an
||z (v)||2 =|| |z (f,-)||2 dt =
0
« T 2 1 _ e-2*T b2
= t\«e-kt )2 dt < b2 —-------< — < b2.
’ 2\ 2\
Consequently, ||z (',')||e D (t) for all 0 < t < T, i. e. D (t ),0< t < T is weakly invariant concerning the problem (l)-(3). Theorem 3 is proved.
References:
1. Egorov A. I. The optimal control of the heat and diffusion processes. Nauka Moscov, 1978 (in Russian).
2. Feuer A., Heymann M. П — invariance in control systems with bounded controls. J. Math.Anal.And Appl.53, no.2 (1976), 266-276.
3. Tukhtasinov M., Kh.Ya. Mustapokulov. Invariant sets in systems with distributed parameters. Theses of the reports of the International Russian-Balkarsky symposium. Nalchik city. 2010. 232-233. (in Russian).
Pirmatov Shamshod Turgunboevich, Head of Higher mathematics department, candidate of physico-mathematical sciences, an associate professor of Tashkent state technical university named after Abu Raikhon Beruni E-mail: shamshod@rambler.ru
Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace
Abstract: The spectral decomposition connected with self-conjugate expansion are considered the operator Laplace in N of dimensional area. It is proved that if spectral decomposition of any function in some point is summarized by Riesz’s means, its average value about a in the specified point possesses the generalized continuity.
Keywords: eigenfuction, eigenvalues, spectral expansion, local bounded variation, Riesz’s means, generalized continuity, summability, operator Laplace.
Introduction. The question on sufficient conditions at which performance it is possible to approximate function by its spectral decomposition connected with self-interfaced expansion of the elliptic operator, by present time is well studied and in detail shined in the mathematical literature as in our country, and abroad. Recently interest to these problems has noticeably increased, and the delicate questions connected with spectral decomposition of rough functions have undergone to research more.
From the mathematical literature well-known the examples showing, that spectral decomposition can converge even in those points where decomposed function has break so the usual requirement of smoothness is not necessary. However in all these examples functions in some the generalized sense nevertheless is continuous in a considered point. This fact for the spectral decomposition ad equating to self-interfaced expansion of operator Laplace in any multivariate area for the first time has been established in E. C. Titchmarsha’s work [1, 255-258]. Corresponding results have been received by them for Riesz means of spectral decomposition in case of when the order of averages is an integer, and also in the assumption, that dimension of considered area is not so great.
Conseder the following orthonormal system of eigenfuc-tions of Laplace operator:
-Au, (x) = A,u, (x),
x efic RN
Define the spectral expansions:
EA(x> f ) =X fkUk(x).
Af. <A
We introduce Riesz means of order s > 0:
E№) = E(z-yj fu= J -j \ dEJ.
Denote
sa f (x)=
R -
|2 \
a>(a, N)
1
\y\<R
1 -1
R2
f (x + y )dy >
where
a>(a,N) = J (l - |yf j dy.
\y\^1
Theoreml. Let f e L2 (D) and s > 0. Suppose that for some x eQ the following ratio is carried out
lim E’J(x) = A.
A—\ /
Then for any a> s — (N — 2)/4 the following equality limS“ f (x) = A
R-^0
is valid.
a
29
Section 6. Mathematics
First we prove some Propositions.
Proposition 1. Let — 1 < s <------. Then
ю
I z(z - 1)s Jv (ajz )dz = r(s + 1)2S+1 a~!~lJv-s_1 (a).
1
For the proofwe can refer, for example, on [3, 717-718]. Set
' A (I +s+1)
ф „ m _ lj (Я-1) Ä~ 2 0 < t < A,
ф A (t, R) _ i t
0, t > A.
Proposition 2. Let l > s —1/2 and s > —1. Then the following equality
(s-1 +1)
КщФa (t,R) = t 2 r(s + 1)2s+1(Wf )-s-1 J1 (Ryft)
we
is valid.
Proof. We have
Yt _t_ ^ s s ___
ФA(t,R) = J tst---(u-1)su~J,+,JR-Jm)tdu =
t
A/
si 1 't I 1 s ,
= ti_i+i J (u -1)1 u~T~T~2 Jl+s+1(RJtu)du.
1
We set V = l +1 + s .Then since l > s —1/2 and s > — 1 have
v-1/4 = I/ 2 + s/2 +1/4 = (I - s + \/2)l2 + s > s , or -1 < s <v/2 -1/4.
In this case the required equality follows from Proposi-tionl.
Q. E.D.
Proposition 3. Let l > s +1/2. The following inequality
|Фa(t,R)| <0(t,R), t > 1,
where
Ф(t, R) = C (R)t ~{l -s"1/2)/2 is valid.
Proof. We have
Ф a (t ,u)\t
s l 1
(l-s-1)
7 -(i+l+s)
j (u - 1)su 2 Jj++i+s [(Thjt )V u ]di
< I u2 2 2 Jl+1+s (ThWt)
du < s l 3
.. .. w
< C(R)t 1 Ju2 2 4 cos(Thjtu)du-
1
_ 1 w s-l _3
= C(R)t 4 J v 2 cos(Ryftv)dv <
-1s-l-1 г -1
< C(R)t 4 J v 2 cos(Thjtv)dv =C(R)t 4.
1
Q. E.D.
Proposition 4. Let g (t) he the function of local bounded variation on the half-line t > 0 and for some ß > 0 the following inequality
Jt ß\dg(t)| <да
gs(A) = J (1 -ffdg(t).
0
If l > ß + s +1/2 then the following equality j (Thjt)-1 Jl (R>Jt )dg (t) =
R
Ю _____
-j (yfA)-1 ~1+sJ,+1+S (Ых)Е : fdX
2 r(s +1);
is valid.
Proof. Consider the following integral
IA =j (yfX)-' +1+s (Ryfl)g! (X)dX'
0
Then
IA = J J1+1JRsU)g! (X)dX =
dX = dX =
= JX 2~2+2JI+I+1(WÄ) J^l-~Xj dg(t)
A _ l_ 1 _s Г X
= JX"2_2+2Jl+1+1(WX) J (X _ tуX~sdg(t)
0 L 0
A A l 1 s
=Jdg(t)J(X_tуX 2_2_2J,+s+i(WX)dx =
0 t
= |фA(t>R)dg(t).
Hence,
IA =|Ф A ( > R)dg (t).
0
Remaind l > s —1/2. In this case, according to Proposition 2,
lim Ф a (t, R) = r(s +1)2 s+1(Th/t )-s-1 J, (Wf)
A—.
Further, since l > s +1/2 according to Proposition 3,
Ю Ю Ю
||0(f,u)\\dg(t) < C(h)jt~((~x~x,2)l2 |dg(t)| < CJt~ß \dg(t) < ю 1 1 1
Hens, we may apply theorem of Lebesgue:
I = lim IA = lim f Ф A (t, R)dg (t) =
0
ю l
= r(s + 1)2s+1 R -s-1J t2 J (Wt )dg (t).
0
Q. E.D.
From this moment we set l = N/2 + a.
Proposition 5. Let l = N/2 + a, where a > —1/2. Then
2lr(l +1)J (Ых)~‘J,(RjÄ)dExf(x) = SaR=(x),
0
and integral converges absolutely and uniformly.
For the proof we can refer on [1, 255-258]. Proposition 6. Set g(A) = Ex f (x). Let ß > N/4. Then
]x~ß\dg (t )| < C||/||.
0
Proposition 7. Let l = N/2 + a,where a> s — (N — 2)/4.
Then
SaRf(x) = 2мR+1~lJ (JÄy-1+!Il+1+s(^X)RJdX.
is valid. Set
Г( s +1)
<
0
30
Necessary conditions of summability of spectral expansion on eigenfuction of the operator laplace
Proof follows from Proposition 2 and 3.
Corollary. Let l = N/2 + a, where a> s — (N — 2)/4 . Then
Кf(x) = 2--J g(XR-=)(JГ'"J„„,(-It)dt.
r(s +1) о
Proposition 8. Let l = N/2 + a, where a > s — (N — 2)/4.
Then
к
Jt-a+i-sVi ji)dt<к.
Proof of the Theoreml follows from Propositions 7, 8. Q. E.D.
References:
1. Titchmarsh E. C. Eigenfunction Expansions Associated with Second Order Differential Equations, part II, Oxford, 1958.
2. Bochner S. Summation of multiple Fourier series by spherical means. Trans. Amer. Math. Soc., Volume 40, 1936.
3. Gradshteyn, Ryzhik, Tables of Integrals and Sums. M.1963.
4. Alimov Sh. A., Ilyin V. A. Condition exactes de convergence uniforme des developpements spectraux et de leurs moyennes de Riesz pour une extension autoadjoint arbitraire de l’operateur de Laplace. Comptes Rendus Acad. Scien. Paris, Serie A, t. 237, 1970.
5. Sogge C. D. Eigenfunction and Bochner-Riesz estimates on manifolds with boundary, Mathematical Research Letter, Volume 9, 2002.
6. Alimov Sh. A. Sets of uniform convergence of Fourier expansions of piecewise smooth functions, J. of Fourier Analysis and Applications, Volume 10 (6), 2004.
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